# Properties

 Label 1155.2.i.b Level 1155 Weight 2 Character orbit 1155.i Analytic conductor 9.223 Analytic rank 0 Dimension 8 CM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1155 = 3 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1155.i (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$9.22272143346$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( \beta_{4} - \beta_{5} ) q^{2}$$ $$-\beta_{3} q^{3}$$ $$+ ( -\beta_{2} + \beta_{3} - \beta_{6} ) q^{4}$$ $$+ \beta_{3} q^{5}$$ $$+ ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{6}$$ $$+ ( \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{7}$$ $$+ ( -\beta_{1} + \beta_{4} ) q^{8}$$ $$- q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( \beta_{4} - \beta_{5} ) q^{2}$$ $$-\beta_{3} q^{3}$$ $$+ ( -\beta_{2} + \beta_{3} - \beta_{6} ) q^{4}$$ $$+ \beta_{3} q^{5}$$ $$+ ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{6}$$ $$+ ( \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{7}$$ $$+ ( -\beta_{1} + \beta_{4} ) q^{8}$$ $$- q^{9}$$ $$+ ( -\beta_{1} - \beta_{5} - \beta_{7} ) q^{10}$$ $$+ ( -3 - \beta_{1} + \beta_{5} ) q^{11}$$ $$+ ( -1 + \beta_{2} - \beta_{6} ) q^{12}$$ $$+ ( -3 \beta_{1} - 3 \beta_{5} ) q^{13}$$ $$+ ( -2 - 2 \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{14}$$ $$+ q^{15}$$ $$+ ( -1 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{16}$$ $$+ ( \beta_{1} + \beta_{5} + 2 \beta_{7} ) q^{17}$$ $$+ ( -\beta_{4} + \beta_{5} ) q^{18}$$ $$+ ( 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{7} ) q^{19}$$ $$+ ( 1 - \beta_{2} + \beta_{6} ) q^{20}$$ $$+ ( 3 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{21}$$ $$+ ( 1 + \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{22}$$ $$-4 q^{23}$$ $$+ \beta_{7} q^{24}$$ $$- q^{25}$$ $$+ ( 3 - 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{6} ) q^{26}$$ $$+ \beta_{3} q^{27}$$ $$+ ( -\beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{28}$$ $$+ ( 2 \beta_{4} - 2 \beta_{5} ) q^{29}$$ $$+ ( \beta_{4} - \beta_{5} ) q^{30}$$ $$+ ( 4 - 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{6} ) q^{31}$$ $$+ ( -4 \beta_{1} - \beta_{4} + 5 \beta_{5} ) q^{32}$$ $$+ ( -\beta_{1} + 3 \beta_{3} - \beta_{5} ) q^{33}$$ $$+ ( -1 + \beta_{2} + 3 \beta_{3} - \beta_{6} ) q^{34}$$ $$+ ( -3 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{35}$$ $$+ ( \beta_{2} - \beta_{3} + \beta_{6} ) q^{36}$$ $$+ ( -4 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{37}$$ $$+ ( -2 + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{6} ) q^{38}$$ $$+ ( 3 \beta_{1} - 3 \beta_{5} ) q^{39}$$ $$-\beta_{7} q^{40}$$ $$+ ( -2 \beta_{1} - 2 \beta_{5} + 2 \beta_{7} ) q^{41}$$ $$+ ( -1 + \beta_{2} + 3 \beta_{3} - 2 \beta_{6} ) q^{42}$$ $$+ ( -\beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{43}$$ $$+ ( -\beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{44}$$ $$-\beta_{3} q^{45}$$ $$+ ( -4 \beta_{4} + 4 \beta_{5} ) q^{46}$$ $$+ ( -4 + 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{6} ) q^{47}$$ $$+ ( -2 + 2 \beta_{2} + \beta_{3} - 2 \beta_{6} ) q^{48}$$ $$+ ( -4 \beta_{2} + 3 \beta_{3} - 4 \beta_{6} ) q^{49}$$ $$+ ( -\beta_{4} + \beta_{5} ) q^{50}$$ $$+ ( \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{51}$$ $$+ ( 3 \beta_{1} + 3 \beta_{5} + 6 \beta_{7} ) q^{52}$$ $$+ ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{53}$$ $$+ ( -\beta_{1} - \beta_{5} - \beta_{7} ) q^{54}$$ $$+ ( \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{55}$$ $$+ ( 1 - 2 \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{56}$$ $$+ ( -2 \beta_{4} + 2 \beta_{5} ) q^{57}$$ $$+ ( -4 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{58}$$ $$+ ( -4 + 4 \beta_{2} - 6 \beta_{3} - 4 \beta_{6} ) q^{59}$$ $$+ ( -\beta_{2} + \beta_{3} - \beta_{6} ) q^{60}$$ $$+ ( -2 \beta_{1} - 2 \beta_{5} - 4 \beta_{7} ) q^{61}$$ $$+ ( 12 \beta_{1} + 12 \beta_{5} + 8 \beta_{7} ) q^{62}$$ $$+ ( -\beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{63}$$ $$+ ( 4 + \beta_{2} - \beta_{3} + \beta_{6} ) q^{64}$$ $$+ ( -3 \beta_{1} + 3 \beta_{5} ) q^{65}$$ $$+ ( 1 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{66}$$ $$+ ( -2 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{67}$$ $$+ ( -3 \beta_{1} - 3 \beta_{5} ) q^{68}$$ $$+ 4 \beta_{3} q^{69}$$ $$+ ( 1 - \beta_{2} - 3 \beta_{3} + 2 \beta_{6} ) q^{70}$$ $$+ ( -4 \beta_{2} + 4 \beta_{3} - 4 \beta_{6} ) q^{71}$$ $$+ ( \beta_{1} - \beta_{4} ) q^{72}$$ $$+ ( -5 \beta_{1} - 5 \beta_{5} - 4 \beta_{7} ) q^{73}$$ $$+ ( -2 \beta_{1} - 6 \beta_{4} + 8 \beta_{5} ) q^{74}$$ $$+ \beta_{3} q^{75}$$ $$+ ( -4 \beta_{1} - 4 \beta_{5} - 2 \beta_{7} ) q^{76}$$ $$+ ( 3 + \beta_{3} - 3 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{77}$$ $$+ ( -3 - 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{6} ) q^{78}$$ $$+ ( 2 \beta_{1} - 2 \beta_{4} ) q^{79}$$ $$+ ( 2 - 2 \beta_{2} - \beta_{3} + 2 \beta_{6} ) q^{80}$$ $$+ q^{81}$$ $$+ ( 2 - 2 \beta_{2} + 2 \beta_{6} ) q^{82}$$ $$+ ( -\beta_{1} - \beta_{5} ) q^{83}$$ $$+ ( -2 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} ) q^{84}$$ $$+ ( -\beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{85}$$ $$+ ( -3 - \beta_{2} + \beta_{3} - \beta_{6} ) q^{86}$$ $$+ ( 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{7} ) q^{87}$$ $$+ ( -1 + 3 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{6} ) q^{88}$$ $$+ ( -4 + 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{6} ) q^{89}$$ $$+ ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{90}$$ $$+ ( 9 - 6 \beta_{2} - 3 \beta_{3} ) q^{91}$$ $$+ ( 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{6} ) q^{92}$$ $$+ ( -4 - 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{6} ) q^{93}$$ $$+ ( -12 \beta_{1} - 12 \beta_{5} - 8 \beta_{7} ) q^{94}$$ $$+ ( 2 \beta_{4} - 2 \beta_{5} ) q^{95}$$ $$+ ( -5 \beta_{1} - 5 \beta_{5} - \beta_{7} ) q^{96}$$ $$+ ( 4 - 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{6} ) q^{97}$$ $$+ ( -3 \beta_{1} - 4 \beta_{4} + 9 \beta_{5} + \beta_{7} ) q^{98}$$ $$+ ( 3 + \beta_{1} - \beta_{5} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q$$ $$\mathstrut -\mathstrut 8q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$8q$$ $$\mathstrut -\mathstrut 8q^{9}$$ $$\mathstrut -\mathstrut 24q^{11}$$ $$\mathstrut -\mathstrut 20q^{14}$$ $$\mathstrut +\mathstrut 8q^{15}$$ $$\mathstrut -\mathstrut 8q^{16}$$ $$\mathstrut +\mathstrut 8q^{22}$$ $$\mathstrut -\mathstrut 32q^{23}$$ $$\mathstrut -\mathstrut 8q^{25}$$ $$\mathstrut -\mathstrut 32q^{37}$$ $$\mathstrut +\mathstrut 4q^{42}$$ $$\mathstrut -\mathstrut 4q^{56}$$ $$\mathstrut -\mathstrut 32q^{58}$$ $$\mathstrut +\mathstrut 32q^{64}$$ $$\mathstrut -\mathstrut 16q^{67}$$ $$\mathstrut -\mathstrut 4q^{70}$$ $$\mathstrut +\mathstrut 16q^{77}$$ $$\mathstrut -\mathstrut 24q^{78}$$ $$\mathstrut +\mathstrut 8q^{81}$$ $$\mathstrut -\mathstrut 24q^{86}$$ $$\mathstrut -\mathstrut 8q^{88}$$ $$\mathstrut +\mathstrut 48q^{91}$$ $$\mathstrut -\mathstrut 32q^{93}$$ $$\mathstrut +\mathstrut 24q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{3}$$ $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4} + \zeta_{24}^{2}$$ $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{6}$$ $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}$$ $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{5} + \zeta_{24}$$ $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{4} + \zeta_{24}^{2}$$ $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5}$$
 $$1$$ $$=$$ $$\beta_0$$ $$\zeta_{24}$$ $$=$$ $$($$$$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}$$$$)/2$$ $$\zeta_{24}^{2}$$ $$=$$ $$($$$$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{2}$$$$)/2$$ $$\zeta_{24}^{3}$$ $$=$$ $$\beta_{1}$$ $$\zeta_{24}^{4}$$ $$=$$ $$($$$$-$$$$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{2}$$$$)/2$$ $$\zeta_{24}^{5}$$ $$=$$ $$($$$$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}$$$$)/2$$ $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{3}$$ $$\zeta_{24}^{7}$$ $$=$$ $$($$$$-$$$$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}$$$$)/2$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$232$$ $$386$$ $$661$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 −0.965926 − 0.258819i 0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 − 0.965926i 0.258819 + 0.965926i −0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 + 0.258819i
1.93185i 1.00000i −1.73205 1.00000i −1.93185 0.189469 2.63896i 0.517638i −1.00000 1.93185
76.2 1.93185i 1.00000i −1.73205 1.00000i 1.93185 −0.189469 2.63896i 0.517638i −1.00000 −1.93185
76.3 0.517638i 1.00000i 1.73205 1.00000i −0.517638 −2.63896 + 0.189469i 1.93185i −1.00000 0.517638
76.4 0.517638i 1.00000i 1.73205 1.00000i 0.517638 2.63896 + 0.189469i 1.93185i −1.00000 −0.517638
76.5 0.517638i 1.00000i 1.73205 1.00000i 0.517638 2.63896 0.189469i 1.93185i −1.00000 −0.517638
76.6 0.517638i 1.00000i 1.73205 1.00000i −0.517638 −2.63896 0.189469i 1.93185i −1.00000 0.517638
76.7 1.93185i 1.00000i −1.73205 1.00000i 1.93185 −0.189469 + 2.63896i 0.517638i −1.00000 −1.93185
76.8 1.93185i 1.00000i −1.73205 1.00000i −1.93185 0.189469 + 2.63896i 0.517638i −1.00000 1.93185
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 76.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.b Odd 1 no
11.b Odd 1 no
77.b Even 1 no

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{4}$$ $$\mathstrut +\mathstrut 4 T_{2}^{2}$$ $$\mathstrut +\mathstrut 1$$ acting on $$S_{2}^{\mathrm{new}}(1155, [\chi])$$.